How to rig a "cartoonish" style mouth that doesn't affect the rest of the face?
I think your topology is not good for a head, basically you've kept the topology of a UV sphere, it won't help to deform properly, what you need to do is create a kind of topology that follows the face features (found on google, from 3D artist Israel Audelo &quot;izzy&quot;):

See how the topology rotates concentrically around the mouth. As your character is even simpler and the expression more exagarate you should do the same with no more edge loops than necessary: